An Improved Lower Bound for \(g^{(4)}(18)\)

The cardinality of the minimal pairwise balanced designs on \( v \) elements with largest block size \( k \) is denoted by \( g^{(k)}(v) \). It is known that

\[
30 \leq g^{(4)}(18) \leq 33.
\]

In this note, we show that

\[
31 \leq g^{(4)}(18).
\]